Shape of a SelfAvoiding Walk or Polymer Chain
Abstract
If p_{n}(r) is the probability density of selfavoiding walks of m steps which reach the point r it is proved rigorously that the generating function P(θ,r)= ∑ n=1∞ p_{n}(r)exp(nθ) decays exponentially with r. This result is used to derive restrictions on the form of the distribution p_{n}(r). In particular it is argued that if, for large n the distribution in d dimensions approaches a limiting shape, R_{n}^{d}F(r/R_{n}), where the scaling length R_{n}, which measures the mean endtoend distance, varies as r_{0}n^{v}, then the shape factor has the form F(y)=A(y)exp(y^{δ}), where δ=1/(1—v) and where A(y) does not vary exponentially fast for large y. Accepting the values v_{2}=¾ and v_{3}=⅗ for d=2 and 3, as suggested originally by Flory and since supported by numerical and theoretical calculations, yields δ_{2}=4 and δ_{3}=2 1/2 so that F(y) has the form conjectured recently by Domb et al. on the basis of the numerical analysis of finite lattice chains.
 Publication:

Journal of Chemical Physics
 Pub Date:
 January 1966
 DOI:
 10.1063/1.1726734
 Bibcode:
 1966JChPh..44..616F